Regular hexagon and its properties. How to Draw a Hexagonal Geometric Pattern in Adobe Illustrator

Geometric patterns are very popular lately. In today's lesson, we will learn how to create one of these patterns. Using transition, typography and trendy colors, we will create a pattern that you can use in web and print design.

Result

Step 2
Draw another hexagon, smaller this time - select a radius in 20pt.

2. Transition between hexagons

Step 1
Select both hexagons and align them to the center (vertically and horizontally). Using the tool Blend/Transition (W), select both hexagons and make them transition to 6 Steps. To make it easier to see, change the color of the shapes before the transition.

3. Divide into sections

Step 1
Tool Line Segment (\) draw a line crossing the hexagons in the center from the leftmost corner to the rightmost. Draw two more lines crossing the hexagons centered from opposite corners.

4. Painting over the sections

Step 1
Before we start painting over the sections, let's define the palette. Here is the palette from the example:

• Blue: C 65 M 23 Y 35 K 0
• Beige: C 13 M 13 Y 30 K 0
• Peach: C 0 M 32 Y 54 K 0
• Light pink: C 0 M 64 Y 42 K 0
• Dark pink: C 30 M 79 Y 36 K 4

The example immediately used CMYK mode so that the pattern could be printed without modification.

5. Finishing touches and pattern

Step 1
Group (Control-G) all sections and hexagons after you are done with their coloring. Copy (Control-C) And Paste (Control-V) group of hexagons. Let's name the original group Hexagon A, and its copy Hexagon B. Align the groups.

Step 2
Apply Linear Gradient to the group Hexagon B. In the palette Gradient / Gradient specify a fill from purple ( C60 M86 Y45 K42) to cream color ( C0 M13 Y57 K0).

The topic of polygons is covered in the school curriculum, but they do not pay enough attention to it. Meanwhile, it is interesting, and this is especially true of a regular hexagon or hexagon - after all, many natural objects have this shape. These include honeycombs and more. This form is very well applied in practice.

Definition and construction

A regular hexagon is a plane figure that has six sides equal in length and the same number of equal angles.

If we recall the formula for the sum of the angles of a polygon

it turns out that in this figure it is equal to 720 °. Well, since all the angles of the figure are equal, it is easy to calculate that each of them is equal to 120 °.

Drawing a hexagon is very simple, all you need is a compass and a ruler.

The step-by-step instruction will look like this:

If desired, you can do without a line by drawing five circles of equal radius.

The figure thus obtained will be a regular hexagon, and this can be proved below.

Properties are simple and interesting

To understand the properties of a regular hexagon, it makes sense to break it into six triangles:

This will help in the future to more clearly display its properties, the main of which are:

1. circumscribed circle diameter;
2. diameter of the inscribed circle;
3. square;
4. perimeter.

The circumscribed circle and the possibility of construction

It is possible to describe a circle around a hexagon, and moreover, only one. Since this figure is correct, you can do it quite simply: draw a bisector from two adjacent angles inside. They intersect at point O, and together with the side between them form a triangle.

The angles between the side of the hexagon and the bisectors will be 60° each, so we can definitely say that a triangle, for example, AOB, is isosceles. And since the third angle will also be equal to 60 °, it is also equilateral. It follows that the segments OA and OB are equal, which means that they can serve as the radius of the circle.

After that, you can go to the next side, and also draw a bisector from the angle at point C. It will turn out another equilateral triangle, and side AB will be common to two at once, and OS will be the next radius through which the same circle goes. There will be six such triangles in total, and they will have a common vertex at point O. It turns out that it will be possible to describe the circle, and it is only one, and its radius is equal to the side of the hexagon:

That is why it is possible to build this figure with the help of a compass and a ruler.

Well, the area of ​​\u200b\u200bthis circle will be standard:

Inscribed circle

The center of the circumscribed circle coincides with the center of the inscribed one. To verify this, we can draw perpendiculars from the point O to the sides of the hexagon. They will be the heights of those triangles that make up the hexagon. And in an isosceles triangle, the height is the median with respect to the side on which it rests. Thus, this height is nothing but the perpendicular bisector, which is the radius of the inscribed circle.

The height of an equilateral triangle is calculated simply:

h²=a²-(a/2)²= a²3/4, h=a(√3)/2

And since R=a and r=h, it turns out that

r=R(√3)/2.

Thus, the inscribed circle passes through the centers of the sides of a regular hexagon.

Its area will be:

S=3πa²/4,

that is, three-quarters of that described.

Perimeter and area

Everything is clear with the perimeter, this is the sum of the lengths of the sides:

P=6a, or P=6R

But the area will be equal to the sum of all six triangles into which the hexagon can be divided. Since the area of ​​a triangle is calculated as half the product of the base and the height, then:

S \u003d 6 (a / 2) (a (√3) / 2) \u003d 6a² (√3) / 4 \u003d 3a² (√3) / 2 or

S=3R²(√3)/2

Those who wish to calculate this area through the radius of the inscribed circle can be done like this:

S=3(2r/√3)²(√3)/2=r²(2√3)

Entertaining constructions

A triangle can be inscribed in a hexagon, the sides of which will connect the vertices through one:

There will be two of them in total, and their imposition on each other will give the Star of David. Each of these triangles is equilateral. This is easy to verify. If you look at the AC side, then it belongs to two triangles at once - BAC and AEC. If in the first of them AB \u003d BC, and the angle between them is 120 °, then each of the remaining ones will be 30 °. From this we can draw logical conclusions:

1. The height of ABC from vertex B will be equal to half the side of the hexagon, since sin30°=1/2. Those who wish to verify this can be advised to recalculate according to the Pythagorean theorem, it fits here perfectly.
2. The AC side will be equal to two radii of the inscribed circle, which is again calculated using the same theorem. That is, AC=2(a(√3)/2)=a(√3).
3. Triangles ABC, CDE and AEF are equal in two sides and the angle between them, and hence the equality of sides AC, CE and EA follows.

Intersecting with each other, the triangles form a new hexagon, and it is also regular. It's easy to prove:

Thus, the figure meets the signs of a regular hexagon - it has six equal sides and angles. From the equality of triangles at the vertices, it is easy to deduce the length of the side of the new hexagon:

d=а(√3)/3

It will also be the radius of the circle described around it. The radius of the inscribed will be half the side of the large hexagon, which was proved when considering the triangle ABC. Its height is exactly half of the side, therefore, the second half is the radius of the circle inscribed in the small hexagon:

r₂=а/2

S=(3(√3)/2)(а(√3)/3)²=а(√3)/2

It turns out that the area of ​​​​the hexagon inside the star of David is three times smaller than that of the large one in which the star is inscribed.

From theory to practice

The properties of the hexagon are very actively used both in nature and in various fields of human activity. First of all, this applies to bolts and nuts - the hats of the first and second are nothing more than a regular hexagon, if you do not take into account the chamfers. The size of wrenches corresponds to the diameter of the inscribed circle - that is, the distance between opposite faces.

Has found its application and hexagonal tiles. It is much less common than a quadrangular one, but it is more convenient to lay it: three tiles meet at one point, not four. Compositions can be very interesting:

Concrete paving slabs are also produced.

The prevalence of the hexagon in nature is explained simply. Thus, it is easiest to fit circles and balls tightly on a plane if they have the same diameter. Because of this, honeycombs have such a shape.

Content:

A regular hexagon, also called a perfect hexagon, has six equal sides and six equal angles. You can draw a hexagon with a tape measure and a protractor, a rough hexagon with a round object and a ruler, or an even rougher hexagon with just a pencil and a little intuition. If you want to know how to draw a hexagon in different ways, just read on.

Steps

1 Draw a perfect hexagon with a compass

1. 1 Draw a circle using a compass. Insert the pencil into the compass. Expand the compass to the desired width of the radius of your circle. The radius can be from a couple to tens of centimeters wide. Next, put a compass with a pencil on paper and draw a circle.
   • Sometimes it's easier to draw the half of the circle first and then the other half.
2. 2 Move the compass needle to the edge of the circle. Put it on top of the circle. Do not change the angle and position of the compass.
3. 3 Make a small pencil mark on the edge of the circle. Make it distinct, but not too dark, as you will erase it later. Remember to save the angle you set for the compass.
4. 4 Move the compass needle to the mark you just made. Set the needle straight on the mark.
5. 5 Make another mark with a pencil on the edge of the circle. Thus, you will make a second mark at a certain distance from the first mark. Keep moving in one direction.
6. 6 Make four more marks in the same way. You must return back to the original mark. If not, then most likely the angle at which you held the compass and made the marks has changed. Perhaps this happened due to the fact that you squeezed it too hard or, on the contrary, loosened it a little.
7. 7 Connect the marks with a ruler. The six places where your marks intersect with the edge of the circle are the six vertices of the hexagon. Using a ruler and pencil, draw straight lines connecting adjacent marks.
8. 8 Erase both the circle and the marks on the edges of the circle and any other marks you have made. After you have erased all your guide lines, your perfect hexagon should be ready.

2 Draw a rough hexagon with a round object and a ruler

1. 1 Circle the rim of the glass with a pencil. This way you will draw a circle. It is very important to draw with a pencil, because later you will need to erase all the auxiliary lines. You can also circle an upside down glass, jar, or anything else that has a round base.
2. 2 Draw horizontal lines across the center of your circle. You can use a ruler, a book, anything with a straight edge. If you do have a ruler, you can mark the middle by calculating the vertical length of the circle and dividing it in half.
3. 3 Draw an "X" over the half circle, dividing it into six equal sections. Since you've already drawn a line through the middle of the circle, the X must be wider than it is tall for the parts to be equal. Imagine that you are dividing a pizza into six pieces.
4. 4 Make triangles from each section. To do this, use your ruler to draw a straight line under the curved portion of each section, connecting it with the other two lines to form a triangle. Do this with the remaining five sections. Think of it like making the crust around your pizza slices.
5. 5 Erase all auxiliary lines. The guide lines include your circle, the three lines that divided your circle into sections, and any other marks you made along the way.

3 Draw a rough hexagon with one pencil

1. 1 Draw a horizontal line. To draw a straight line without a ruler, simply draw the start and end point of your horizontal line. Then place the pencil at the starting point and extend the line to the end. The length of this line can be only a couple of centimeters.
2. 2 Draw two diagonal lines from the ends of the horizontal one. The diagonal line on the left side should point outward in the same way as the diagonal line on the right. You can imagine that these lines form a 120 degree angle with respect to the horizontal line.
3. 3 Draw two more horizontal lines coming from the first horizontal lines drawn inwards. This will create a mirror image of the first two diagonal lines. The bottom left line should be a reflection of the top left line, and the bottom right line should be a reflection of the top right line. While the top horizontal lines should face outward, the bottom lines should look inward at the base.
4. 4 Draw another horizontal line, connecting the bottom two diagonal lines. This way you will draw the base for your hexagon. Ideally, this line should be parallel to the top horizontal line. Here you have completed your hexagon.

• Pencil and compasses should be sharp to minimize errors from marks that are too wide.
• When using the compass method, if you connected every mark instead of all six, you get an equilateral triangle.

Warnings

• The compass is a rather sharp object, be very careful with it.

Principle of operation

• Each method will help draw a hexagon formed by six equilateral triangles with a radius equal to the length of all sides. The six drawn radii are the same length and all the lines to create the hexagon are also the same length, since the width of the compass did not change. Due to the fact that the six triangles are equilateral, the angles between their vertices are 60 degrees.

What will you need

• Paper
• Pencil
• Ruler
• Pair of compasses
• Something that can be placed under the paper to keep the compass needle from slipping.
• Eraser

Construction of a regular hexagon inscribed in a circle. The construction of a hexagon is based on the fact that its side is equal to the radius of the circumscribed circle. Therefore, to build, it is enough to divide the circle into six equal parts and connect the found points to each other (Fig. 60, a).

A regular hexagon can be constructed using a T-square and a 30X60° square. To perform this construction, we take the horizontal diameter of the circle as the bisector of angles 1 and 4 (Fig. 60, b), build sides 1-6, 4-3, 4-5 and 7-2, after which we draw sides 5-6 and 3- 2.

Construction of an equilateral triangle inscribed in a circle. The vertices of such a triangle can be constructed using a compass and a square with angles of 30 and 60 °, or only one compass.

Consider two ways to construct an equilateral triangle inscribed in a circle.

First way(Fig. 61, a) is based on the fact that all three angles of the triangle 7, 2, 3 each contain 60 °, and the vertical line drawn through the point 7 is both the height and the bisector of angle 1. Since the angle 0-1- 2 is equal to 30°, then to find the side

1-2, it is enough to build an angle of 30 ° at point 1 and side 0-1. To do this, set the T-square and square as shown in the figure, draw a line 1-2, which will be one of the sides of the desired triangle. To build side 2-3, set the T-square to the position shown by the dashed lines, and draw a straight line through point 2, which will define the third vertex of the triangle.

Second way is based on the fact that if you build a regular hexagon inscribed in a circle, and then connect its vertices through one, you get an equilateral triangle.

To construct a triangle (Fig. 61, b), we mark a vertex-point 1 on the diameter and draw a diametrical line 1-4. Further, from point 4 with a radius equal to D / 2, we describe the arc until it intersects with the circle at points 3 and 2. The resulting points will be two other vertices of the desired triangle.

Construction of a square inscribed in a circle. This construction can be done using a square and a compass.

The first method is based on the fact that the diagonals of the square intersect in the center of the circumscribed circle and are inclined to its axes at an angle of 45°. Based on this, we install a T-square and a square with angles of 45 ° as shown in Fig. 62, a, and mark points 1 and 3. Further, through these points, we draw the horizontal sides of the square 4-1 and 3-2 with the help of a T-square. Then, using a T-square along the leg of the square, we draw the vertical sides of the square 1-2 and 4-3.

The second method is based on the fact that the vertices of the square bisect the arcs of the circle enclosed between the ends of the diameter (Fig. 62, b). We mark points A, B and C at the ends of two mutually perpendicular diameters, and from them with a radius y we describe the arcs until they intersect.

Further, through the points of intersection of the arcs, we draw auxiliary lines, marked on the figure with solid lines. Their points of intersection with the circle will define vertices 1 and 3; 4 and 2. The vertices of the desired square obtained in this way are connected in series with each other.

Construction of a regular pentagon inscribed in a circle.

To inscribe a regular pentagon in a circle (Fig. 63), we make the following constructions.

We mark point 1 on the circle and take it as one of the vertices of the pentagon. Divide segment AO in half. To do this, with the radius AO from point A, we describe the arc to the intersection with the circle at points M and B. Connecting these points with a straight line, we get the point K, which we then connect to point 1. With a radius equal to the segment A7, we describe the arc from point K to the intersection with the diametrical line AO ​​at point H. Connecting point 1 with point H, we get the side of the pentagon. Then, with a compass opening equal to the segment 1H, describing the arc from vertex 1 to the intersection with the circle, we find vertices 2 and 5. Having made notches from vertices 2 and 5 with the same compass opening, we obtain the remaining vertices 3 and 4. We connect the found points sequentially with each other.

Construction of a regular pentagon given its side.

To construct a regular pentagon along its given side (Fig. 64), we divide the segment AB into six equal parts. From points A and B with radius AB we describe arcs, the intersection of which will give point K. Through this point and division 3 on the line AB we draw a vertical line.

We get the point 1-vertex of the pentagon. Then, with a radius equal to AB, from point 1 we describe the arc to the intersection with the arcs previously drawn from points A and B. The intersection points of the arcs determine the vertices of the pentagon 2 and 5. We connect the found vertices in series with each other.

Construction of a regular heptagon inscribed in a circle.

Let a circle of diameter D be given; you need to inscribe a regular heptagon into it (Fig. 65). Divide the vertical diameter of the circle into seven equal parts. From point 7 with a radius equal to the diameter of the circle D, we describe the arc until it intersects with the continuation of the horizontal diameter at point F. Point F is called the pole of the polygon. Taking point VII as one of the vertices of the heptagon, we draw rays from the pole F through even divisions of the vertical diameter, the intersection of which with the circle will determine the vertices VI, V and IV of the heptagon. To obtain vertices / - // - /// from points IV, V and VI, we draw horizontal lines until they intersect with the circle. We connect the found vertices in series with each other. The heptagon can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.

The above method is suitable for constructing regular polygons with any number of sides.

The division of a circle into any number of equal parts can also be done using the data in Table. 2, which shows the coefficients that make it possible to determine the dimensions of the sides of regular inscribed polygons.

A regular circumscribed triangle is constructed as follows(Figure 38). From the center of a given circle of radius R1 draw a circle with a radius R2 = 2R1 and divide it into three equal parts. division points A, B, C are the vertices of a regular triangle circumscribed about a circle of radius R1 .

Figure 38

Regular circumscribed quadrilateral (square) can be built using a compass and a ruler (Figure 39). Two mutually perpendicular diameters are drawn in a given circle. Taking the points of intersection of the diameters with the circle as centers, the radius of the circle R describe arcs until their mutual intersection at points A, B, C, D . points A , B , C , D and are the vertices of the square circumscribed about the given circle.

Figure 39

To construct a regular circumscribed hexagon you must first build the vertices of the described square in the manner indicated above (Figure 40, a). Simultaneously with the definition of the vertices of the square, a given circle of radius R divided into six equal parts at points 1, 2, 3, 4, 5, 6 and draw the vertical sides of the square. Passing through the division points of the circle 2–5 And 3–6 straight lines until they intersect with the vertical sides of the square (Figure 40, b), get vertices A, B, D, E circumscribed regular hexagon.

Figure 40

Other peaks C And F defined by an arc of a circle of radius OA, which is drawn until it intersects with the continuation of the vertical diameter of the given circle.
3 PAIRINGS